Some geometric properties of certain toric varieties and ...1577/fulltext.pdftoric variety. 7. 1. BRUHAT-HIBI TORIC VARIETIES 8 For Lbeing the Bruhat poset of Schubert varieties in

SOME GEOMETRIC PROPERTIES

OF

CERTAIN TORIC VARIETIES AND SCHUBERT VARIETIES

A dissertation presented

by

Justin Allen Brown

to

The Department of Mathematics

In partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in the field of

Mathematics

Northeastern University

Boston, Massachusetts

April, 2009

1

2

SOME GEOMETRIC PROPERTIES

OF

CERTAIN TORIC VARIETIES AND SCHUBERT VARIETIES

by

Justin Allen Brown

ABSTRACT OF DISSERTATION

Submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy in Mathematics

in the Graduate School of Arts and Sciences of

Northeastern University, April, 2009

Abstract

This thesis has three distinct chapters: Bruhat-Hibi toric varieties, Gorenstein

Schubert varieties in a minuscule G/P , and Wahl’s conjecture for a minuscule G/P .

We begin with a study of toric varieties associated to Bruhat lattices for a minuscule

G/P . Our main result is a combinatorial characterization of the singular loci of

these toric varieties. In the next chapter, we are concerned with Schubert varieties

in a minuscule G/P , for which we give a combinatorial characterization for them to

be arithmetically Gorenstein. In the last chapter, we prove Wahl’s conjecture for a

minuscule G/P .

3

Acknowledgment

I would like to thank my advisor, V. Lakshmibai, for sharing with me her deep

understanding of the mathematical world. For that, and much more, I will be for-

ever grateful. I also thank all of my professors and fellow students at Northeastern

University; collectively, I have learned a great deal from you.

I owe a very special thanks to my wife, Jody - for all of your patience, support,

and general wonderfulness.

4

Contents

Abstract 3

Acknowledgment 4

Introduction 7

1. Bruhat-Hibi toric varieties 7

2. Arithmetically Gorenstein Schubert varieties 9

3. Wahl’s conjecture for a minuscule G/P 11

Chapter 1. Bruhat-Hibi toric varieties 13

1. Affine toric varieties 13

2. Distributive lattices 14

3. The variety X(L) 17

4. Grid lattices and minuscule lattices 21

5. Results on a lattice with JIGL 28

6. Singular locus of X(L) 33

7. Divisors and line bundles 42

8. Young lattices 48

9. A counter example 52

10. Multiplicity formulae for G-H toric varieties 55

Chapter 2. Arithmetically Gorenstein Schubert varieties in a minuscule G/P 67

1. Hodge algebras 67

2. Arithmetically Gorenstein Schubert varieties in a Grassmannian 70

5

CONTENTS 6

3. The lattice of Schubert varieties in an orthogonal Grassmannian 73

4. Arithmetically Gorenstein Schubert varieties in an orthogonal

Grassmannian 76

5. The exceptional groups 83

6. The arithmetically Gorenstein property for the flag variety 87

Chapter 3. Wahl’s conjecture for a minuscule G/P 91

1. Frobenius splittings 91

2. Splittings and blow-ups 93

3. Steps leading to a proof of LMP-conjecture for a minuscule G/P 94

4. Steps leading to the determination of ordP/Bσ 96

5. The minuscule SO(2n)/P1 102

6. Exceptional group E6 104

7. Exceptional group E7 109

8. The remaining minuscule G/P ’s 113

Bibliography 122

Introduction

The results of this thesis may be put into the following three headings: Bruhat-

Hibi toric varieties, Gorenstein Schubert varieties in a minuscule G/P , and Wahl’s

conjecture for a minuscule G/P . Accordingly, the results are organized in three

chapters, one for each of the three headings. We now give a brief description of the

results in each of the three chapters.

1. Bruhat-Hibi toric varieties

This thesis project began with a conjecture of Lakshmibai and Gonciulea in [13]

concerning the singular loci of toric degenerations of cones over Grassmannians. Toric

varieties arising in this setting can be realized as the vanishing set of binomials given

by the join and meet relations on a distributive lattice. Specifically, let K denote the

base field which we assume to be algebraically closed of arbitrary characteristic. Given

a distributive lattice L, let X(L) denote the affine variety in A#L whose vanishing

ideal is generated by the binomials XτXϕ − Xτ∨ϕXτ∧ϕ in the polynomial algebra

K[Xα, α ∈ L] (here, τ ∨ ϕ (resp. τ ∧ ϕ) denotes the join - the smallest element of L

greater than both τ, ϕ (resp. the meet - the largest element of L smaller than both

τ, ϕ)). These varieties were extensively studied by Hibi in [14] where Hibi proves that

X(L) is a normal variety. On the other hand, Eisenbud-Sturmfels show in [9] that

a binomial prime ideal is toric (here, “toric ideal” is in the sense of [33]). Thus one

obtains that X(L) is a normal toric variety. We shall refer to such a X (L) as a Hibi

toric variety.

7

1. BRUHAT-HIBI TORIC VARIETIES 8

For L being the Bruhat poset of Schubert varieties in a minuscule G/P , it is

shown in [12] that X(L) flatly deforms to Ĝ/P (the cone over G/P ), i.e., there

exists a flat family over A1 with Ĝ/P as the generic fiber and X(L) as the special

fiber. As mentioned above, the authors in [13] make a conjecture on the singular

locus of X(L) for L a Bruhat poset of Schubert varieties in the Grassmannian. (The

statement of the conjecture is precisely the “Main Result” below.) The sufficiency part

of the conjecture was proven in [13], using the Jacobian criterion for smoothness. The

necessary part of the conjecture has been proven in [1], using certain desingularization

techniques while relating these toric varieties to mirror symmetry.

Neither of [1, 13] brings out the relationship between the singularities of X(L) and

the combinatorics of the polyhedral cone associated to X(L). Lakshmibai suggested

we relate the singularities of X(L) and the combinatorics of the associated polyhedral

cone. The study of such a relationship led us to consider more general distributive

lattices which we call “lattices with JIGL,” namely, distributive lattices such that the

partially ordered subset of join irreducible elements (see Definition 2.5) forms a grid

lattice (a finite sublattice of Z×Z). In §4, we show that all minuscule lattices are in

fact lattices with JIGL (by a minuscule lattice, we mean the Bruhat poset of Schubert

varieties in a minuscule G/P ). For such a lattice L, we call X(L) a Bruhat-Hibi toric

variety.

The goal of the first chapter of this thesis is to study Hibi toric varieties for which

the associated distributive lattice is a lattice with JIGL. The main result gives the

irreducible components of the singular loci of such Hibi toric varieties. This is the

culmination of §5 and §6. We give the statement of the main result here (see Theorem

6.15):

2. ARITHMETICALLY GORENSTEIN SCHUBERT VARIETIES 9

Main Result. Let L be a distributive lattice with JIGL. Then

Sing X(L) =⋃

(α,β)

Zα,β

where (α, β) is an (unordered) pair of incomparable join-meet irreducible elements

in L, and Zα,β = {P ∈ X(L) ⊂ A#L | P (θ) = 0, ∀θ ∈ [α ∧ β, α ∨ β]}. (Here, P (θ)

represents the θ-coordinate of P ∈ A#L for θ ∈ L.)

As a consequence, we obtain a description of the singular locus of X(L) in terms of

the faces of the polyhedral cone associated to X(L); further, we obtain that Sing X(L)

is pure of codimension 3, with generic singularities being of cone type.

In §7, we study the divisors and line bundles of X(L) where L is a lattice with

JIGL. In §8, we show that “Young lattices” are also lattices with JIGL, therefore

the main result above extends to even more Hibi toric varieties of interest (Hibi toric

varieties associated to a Young lattice also appear in [12] as toric degenerations of

the cone over partial flag varieties).

The reader might wonder if the main result above could extend to X(L), where

L is any lattice other than a lattice with JIGL. To answer this question, we give

a counter example in §9. The example uses a lattice for which the poset of join

irreducibles is a sublattice of Z3. We show that the singular locus is not characterized

as in the main result above.

Returning to the original case of toric degenerations of the cone over the Grass-

mannian, we conclude this chapter by giving multiplicity formulae at specific points

of X(L) when L is the well known distributive lattice Id,n (see §10).

2. Arithmetically Gorenstein Schubert varieties

In the next chapter, we are concerned with the problem of characterizing the

arithmetically Gorenstein Schubert varieties in a minuscule G/P (G a semisimple

2. ARITHMETICALLY GORENSTEIN SCHUBERT VARIETIES 10

algebraic group and P a minuscule parabolic subgroup). Let W be the Weyl group

and T a maximal torus. For w ∈ W , we denote by X(w) the Schubert variety in G/P

associated to the T -fixed point wP , i.e., X(w) is the Zariski closure of BwP in G/P ,

with the canonical reduced scheme structure.

While all Schubert varieties are Cohen-Macaulay (cf.[29]), not all of the Schubert

varieties are smooth; one has (thanks to the works of several mathematicians during

1980’s and 1990’s) a complete classification of smooth Schubert varieties (see [2] for

a detailed account on this). The Gorenstein property is a geometric property in

between Cohen-Macaulayness and smoothness properties.

The problem of classifying the Gorenstein Schubert varieties is an open problem.

Recently, Woo and Yong (cf. [36]) have given a characterization of the (geometrically)

Gorenstein Schubert varieties in the flag variety SL(n)/B. As a consequence, one

obtains a combinatorial characterization of the Gorenstein Schubert varieties in the

Grassmannian. The characterization can be described as follows. Let w be an element

in the Weyl group of G/P , then w is associated to a Young diagram. Then X(w) is

Gorenstein if and only if the outer corners of the Young diagram associated to w lie

on the same anti-diagonal.

In Chapter 2, we begin by proving a stronger result than that of [36] (see also

[34]), namely, while in [34, 36], the authors give a characterization of Gorenstein

Schubert varieties, we give a characterization of the Gorenstein property even for the

cones over Schubert varieties. As a consequence, it turns out that a Schubert variety

in the Grassmannian is arithmetically Gorenstein (for the Plücker embedding) if and

only if it is geometrically Gorenstein.

Next, we extend the combinatorial characterization of [36] to arithmetically Gor-

enstein Schubert varieties in the orthogonal Grassmannian, i.e., G/P for G = SO(m)

and P a maximal parabolic subgroup associated to the right end root if m is odd, or

3. WAHL’S CONJECTURE FOR A MINUSCULE G/P 11

one of the right end roots if m is even. (The indexing of the Dynkin diagram is as in

[5].)

The proofs of these results rely on the fact that the coordinate ring of a Schubert

variety is a “Hodge algebra” in the sense of [7], R(w) having a set of algebra generators

indexed by H(w), the Bruhat poset of Schubert subvarieties of X(w). For X(w) a

Schubert variety in a minuscule G/P , we have that H(w) is a distributive lattice.

Now using a result of Stanley (cf.[31]), we have that R(w) is Gorenstein if and only

if the poset of join-irreducibles of H(w) is a ranked poset (i.e., all maximal chains in

H(w) have the same length).

In the remaining minuscule cases, if G is SP (2n) and P = P1 (the maximal para-

bolic corresponding to the left end root in the Dynkin diagram), then G/P ∼= P2n−1;

all of the Schubert varieties in G/P are smooth (and hence Gorenstein). If G is

SO(2n), and P = P1 (the maximal parabolic corresponding to the left end root),

then one easily checks that the poset of join-irreducibles of H(w) is a ranked poset

for all Schubert varieties X(w) in G/P . Hence all of the Schubert varieties are Gor-

enstein. If G is E6 or E7 and P is a minuscule parabolic (using the above criterion for

Gorenstein property), we have listed the arithmetically Gorenstein Schubert varieties

(for the canonical projective embedding X(w) ↪→ Proj (H0 (G/P, L)), L being the

ample generator of Pic(G/P ) ∼= Z).

3. Wahl’s conjecture for a minuscule G/P

In [35], Wahl conjectured that the “Gaussian map” is surjective for the variety

G/P , G a complex semisimple group and P a parabolic subgroup. To be specific, let

X be a non-singular projective variety over C. For ample line bundles L and M over

X, consider the natural restriction map (called the Gaussian)

H0(X ×X, I∆ ⊗ p∗1L⊗ p∗2M)→ H0(X, Ω1X ⊗ L⊗M)

3. WAHL’S CONJECTURE FOR A MINUSCULE G/P 12

where I∆ denotes the ideal sheaf of the diagonal ∆ in X × X, p1 and p2 the two

projections of X × X on X, and Ω1X the sheaf of differential 1-forms of X. Kumar

proved Wahl’s conjecture in [17], using representation theoretic techniques.

In [19], the authors considered Wahl’s conjecture in positive characteristics, and

observed that Wahl’s conjecture will follow if there exists a Frobenius splitting of

X × X which compatibly splits the diagonal and which has the maximum possible

order of vanishing along the diagonal; this stronger statement was formulated as a

conjecture in [19] which we shall refer to as the LMP-conjecture in the sequel:

LMP-conjecture: For any G/P , there exists a splitting of G/P ×

G/P that compatibly splits the diagonal copy of G/P with maximal

multiplicity.

Subsequently, in [25], Mehta-Parameswaran proved the LMP-conjecture for the Grass-

mannian. Recently, Lakshmibai-Raghavan-Sankaran (cf.[22]) extended the result of

[25] to symplectic and orthogonal Grassmannians.

Our goal in this chapter is to show that the LMP conjecture (and hence Wahl’s

conjecture) holds in all characteristics for a minuscule G/P (of course, if G is the

special orthogonal group SO(m), then one should not allow characteristic 2). The

main philosophy in [25, 22] consists of reducing the LMP-conjecture for a G/P to

the problem of finding a section ϕ ∈ H0(G/B, K−1G/B) (KG/B being the canonical

bundle on G/B) which has maximum possible order of vanishing along P/B. Here,

we further reduce this problem to computing the order of vanishing (along P/B) of

the highest weight vector fd in H0(G/B, L(ωd)), for every fundamental weight ωd of

G.

CHAPTER 1

Bruhat-Hibi toric varieties

The main result of this chapter is the description of the singular locus of Bruhat-

Hibi toric varieties. We do so by isolating a specific property that all minuscule

lattices share. Thus, we actually determine the singular locus of a wider class of

Hibi toric varieties, specifically all varieties X(L) such that L is a lattice with join

irreducibles forming a grid lattice.

1. Affine toric varieties

In this section, we recall some basic definitions concerning affine toric varieties.

Let T = (K∗)m be an m-dimensional torus.

Definition 1.1. (cf. [11], [10]) An equivariant affine embedding of a torus T is

an affine variety X ⊆ Al containing T as a dense open subset and equipped with a

T -action T ×X → X extending the action T × T → T given by multiplication. If in

addition X is normal, then X is called an affine toric variety.

1.2. The cone associated to a toric variety. Let X be an affine toric variety,

and T the embedded torus. Let M be the character group of T , and N the Z-dual

of M . Recall (cf. [11], [10]) that there exists a strongly convex rational polyhedral

cone σ ⊂ NR(= N ⊗Z R) such that

K[X] = K[Sσ],

13

2. DISTRIBUTIVE LATTICES 14

where Sσ is the subsemigroup σ∨ ∩M , σ∨ being the cone in MR dual to σ, namely,

σ∨ = {f ∈ MR | f(v) ≥ 0, v ∈ σ}. Note that Sσ is a finitely generated subsemigroup

in M .

We shall denote X also by Xσ. We may suppose, without loss of generality, that

σ spans NR so that the dimension of σ equals dim NR = dim T . (Here, by dimension

of σ, one means the vector space dimension of the span of σ.)

A face τ of σ is the intersection τ = σ ∩ u⊥ = {v ∈ σ | u (v) = 0} for any u ∈ σ∨.

When τ is a face of a cone σ, we denote τ ≤ σ.

1.3. The distinguished point Pτ . Each face τ determines a (closed) point Pτ

in Xσ, namely, it is the point corresponding to the maximal ideal in K[Sσ] given by

the kernel of eτ : K[Sσ]→ K, where for u ∈ Sσ, we have

eτ (u) =

1, if u ∈ τ⊥,0, otherwise,where τ⊥ denotes {u ∈MR |u(v) = 0,∀v ∈ τ}.

1.4. Orbit decomposition. Let Oτ denote the T -orbit in Xσ through Pτ . We

have the following orbit decomposition in Xσ:

Xσ =⋃θ≤σ

Oθ, Oτ =⋃θ≥τ

Oθ,

and dim τ + dim Oτ = dim Xσ. See [11], [10] for details.

2. Distributive lattices

Let (L,≤) be a poset, i.e, a finite partially ordered set. We shall suppose that L

is bounded, i.e., it has a unique maximal, and a unique minimal element, denoted 1̂

and 0̂ respectively. For µ, λ ∈ L, µ ≤ λ, we shall denote

[µ, λ] := {τ ∈ L, µ ≤ τ ≤ λ}


We shall refer to [µ, λ] as the interval from µ to λ.

Definition 2.1. The ordered pair (λ, µ) is called a cover (and we also say that

λ covers µ or µ is covered by λ) if [µ, λ] = {µ, λ}.

Definition 2.2. A lattice is a partially ordered set (L,≤) such that for every pair

of elements x, y ∈ L, there exist elements x∨ y and x∧ y, called the join, respectively

the meet of x and y, defined by:

x ∨ y ≥ x, x ∨ y ≥ y, and if z ≥ x and z ≥ y, then z ≥ x ∨ y,

x ∧ y ≤ x, x ∧ y ≤ y, and if z ≤ x and z ≤ y, then z ≤ x ∧ y.

It is easy to check that the operations ∨ and ∧ are commutative and associative.

Definition 2.3. A lattice is called distributive if the following identities hold:

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z).

Definition 2.4. Given a lattice L, a subset L′ ⊂ L is called a sublattice of L if

x, y ∈ L′ implies x ∧ y ∈ L′, x ∨ y ∈ L′; L′ is called an embedded sublattice of L if

τ, φ ∈ L, τ ∨ φ, τ ∧ φ ∈ L′ ⇒ τ, φ ∈ L′.

Definition 2.5. An element z of a lattice L is called join-irreducible (respectively

meet-irreducible) if z = x∨ y (respectively z = x∧ y) implies z = x or z = y. The set

of join-irreducible (respectively meet-irreducible) elements of L is denoted by J (L)

(respectively M (L)), or just by J (respectively M) if no confusion is possible.

Definition 2.6. An element in J (L) ∩M (L) is called irreducible.

The following Lemma is easily checked.


Lemma 2.7. With the notations as above, we have

(a) J = {τ ∈ L | there exists at most one cover of the form (τ, λ)}.

(b) M = {τ ∈ L | there exists at most one cover of the form (λ, τ)}.

Definition 2.8. A subset I of a poset P is called an ideal of P if for all x, y ∈ P ,

x ∈ I and y ≤ x imply y ∈ I.

Theorem 2.9 (cf. [3]). Let L be a distributive lattice with 0̂, and P the poset of

its nonzero join-irreducible elements. Then L is isomorphic to the lattice of ideals of

P , by means of the lattice isomorphism

α 7→ Iα := {τ ∈ P | τ ≤ α}, α ∈ L.

We will often use the theorem above to view L as the lattice of ideals of J(L).

Thus the notation Iα as defined above will be used frequently.

Lemma 2.10. Let (τ, λ) be a cover in L. Then Iτ equals Iλ∪̇{β} for some β ∈

J (L).

Proof. Let (τ, λ) be a cover in L, then clearly Iτ ) Iλ. Let H = Iτ \ Iλ, we have

that H is non-empty.

Assume, if possible, that β1, β2 are two distinct elements in H ⊂ J(L). Denote

H1 = Iβ1 ∪ Iλ, H2 = Iβ2 ∪ Iλ. Clearly H1 and H2 are ideals in J(L), thus by Theorem

2.9 they correspond to elements in L, call them φ1 and φ2, respectively. We thus have

φ1, φ2 ≤ τ, φ1, φ2 λ.

Note that we have φ1 6= φ2, and φ1 6= λ 6= φ2. Therefore the interval [λ, τ ] has

cardinality of at least three, implying that τ does not cover λ, a contradiction.

Therefore we have that the cardinality of H is equal to one, and Iτ = Iλ∪̇{β} for

some element β. �

3. THE VARIETY X(L) 17

3. The variety X(L)

Consider the polynomial algebra K[Xα, α ∈ L]; let a(L) be the ideal generated by

{XαXβ−Xα∨βXα∧β, α, β ∈ L}. Then one knows (cf.[14]) that K[Xα, α ∈ L] /a(L) is

a normal domain; in particular, we have that a(L) is a prime ideal. Let X(L) be the

affine variety of the zeroes in K#L of a(L). Then X(L) is an affine normal variety

defined by binomials. On the other hand, by [9], we have that a binomial prime ideal

is toric (here, “toric ideal” is in the sense of [33, Chapter 4]). Hence X(L) is a toric

variety for the action by a suitable torus T , and thus dim X(L) = dim T .

In the sequel, we shall denote R (L) := K[Xα, α ∈ L] /a(L). Further, for α ∈ L,

we shall denote the image of Xα in R (L) by xα.

Definition 3.1. The variety X (L) will be called a Hibi toric variety.

Remark 3.2. An extensive study of X (L) appears first in [14].

Theorem 3.3 (cf. [20]). The dimension of X(L) is equal to #J (L). Further,

dimX (L) equals the cardinality of the set of elements in a maximal chain in (the

graded poset) L.

Proposition 3.4. (cf. [13, Proposition 5.16]) X(L′) is a subvariety of X(L) if

and only if L′ is an embedded sublattice of L.

3.5. Multiplicity of X(L) at the origin. In this subsection, we shall determine

the multiplicity of X(L) at the origin. We first recall some definitions.

Let B be a Z+-graded, finitely generated K-algebra, B = ⊕Bm. Let φm(B) denote

the Hilbert function:

φm(B) = dimK Bm.

Let PB(x) denote the Hilbert polynomial of B; recall that


(1) PB(x) ∈ Q[x],

(2) deg PB(x) = dimProjB = s, say,

(3) the leading coefficient of PB(x) is of the formeBs!

.

Definition 3.6. The number eB is called the degree of the graded ring B, or also

the degree of Proj B.

Theorem 3.7. The degree of K[X (L)] is equal to the number of maximal chains

in L.

Proof. Let a(L) be as above. We begin by putting a monomial order on

K[Xα, α ∈ L]. Consider the reverse partial order on L, and extend it to a total

order, denoted ≤tot, on the variables {Xα, α ∈ L}. We now take the monomial order

defined as follows. For α1 ≤tot . . . ≤tot αr, β1 ≤tot . . . ≤tot βs, we say Xα1 · · ·Xαr ≺

Xβ1 · · ·Xβs if and only if either r < s or r = s and there exists a t < r such that

α1 = β1, . . . , αt = βt, αt+1


Note that for m sufficiently large, the leading term appears in the summation above

only for J of maximal cardinality s. The result follows from this. �

Next we recall multP X, the multiplicity of an algebraic variety X at a point P ∈

X: Let OX,P = (A, m). Let CP be the tangent cone at P , namely CP = Spec A(P ),

where A(P ) = gr(A, m). Then the multiplicity of X at P is defined to be:

multP X = deg Proj A(P ) (= deg A(P )).

Hence (using the notation from §3.5) we obtain, eB = mult0 Spec(B), the multi-

plicity of Spec(B) at the origin.

As a direct consequence of Theorem 3.7, we have:

Theorem 3.8. The multiplicity of X(L) at the origin is equal to the number of

maximal chains in L.

3.9. Cone and dual cone of X (L). As above, denote the poset of join-irreduc-

ibles in L by J . Let I (J) denote the poset of ideals of J . For A ∈ I (J), let mA

denote the monomial

mA :=∏τ∈A

yτ

in the polynomial algebra K[yτ , τ ∈ J (L)]. If α is the element of L such that Iα = A

(cf. Theorem 2.9), then we shall denote mA also by mα. Consider the surjective

algebra map

F : K[Xα, α ∈ L]→ K[mA, A ∈ I (J)], Xα 7→mα.

Theorem 3.10 (cf. [14], [20]). We have an isomorphism K[X (L)] ∼= K[mA, A ∈

I (J)].

Let us denote the torus acting on the toric variety X (L) by T ; by Theorem 3.3,

we have, dim T = #J (L) = d, say. Identifying T with (K∗)d, let {fz, z ∈ J (L)}


denote the standard Z-basis for X(T ), namely, for t = (tz)z∈J(L), fz(t) = tz. Denote

M := X(T ); let N be the Z-dual of M , and {ey, y ∈ J (L)} be the basis of N dual to

{fz, z ∈ J (L)}. For A ∈ I (J), set

fA :=∑z∈A

fz.

Let σ ⊂ NR be the cone such that X (L) = Xσ.

As an immediate consequence of Theorem 3.10, we have

Proposition 3.11. The semigroup Sσ is generated by fA, A ∈ I (J).

Let M(J (L)) be the set of maximal elements in the poset J (L). Let Z(J (L))

denote the set of all covers in the poset J (L).

Proposition 3.12 (cf. [20], Proposition 4.7). The cone σ is generated by

{ez | z ∈M(J (L))} ∪ {ey′ − ey | (y, y′) ∈ Z(J (L))}.

3.13. The sublattice Dτ . We shall concern ourselves just with the closed points

in X (L). So in the sequel, by a point in X (L), we shall mean a closed point. Let τ

be a face of σ, and Pτ the distinguished point (see §1.3).

For a point P ∈ X (L) (identified with a point in A#L), let us denote by P (α),

the αth coordinate of P . Let

Dτ = {α ∈ L | Pτ (α) 6= 0}.

We have the following lemma.

Lemma 3.14 (cf. [20]). Dτ is an embedded sublattice. Conversely, let D be an

embedded sublattice in L. Then D determines a unique face τ ′ of σ such that Dτ ′ = D.

Thus in view of the lemma above, we have a bijection

{faces of σ} bij↔{embedded sublattices of L}.

4. GRID LATTICES AND MINUSCULE LATTICES 21

Proposition 3.15 (cf. [20]). Let τ be a face of σ. Then we have Oτ = X (Dτ ).

4. Grid lattices and minuscule lattices

In this section, we restrict our attention to a specific class of distributive lattices.

Give N× N the lattice structure

(α1, α2) ∧ (β1, β2) = (δ1, δ2), (α1, α2) ∨ (β1, β2) = (γ1, γ2),

where δi = min{αi, βi}, γi = max{αi, βi}.

Definition 4.1. Let J be a finite, distributive sublattice of N × N, such that if

α covers β in J , then α covers β in N× N as well. Then we say J is a grid lattice.

Remark 4.2. For J a grid lattice, we have the following:

(1) J is a distributive lattice.

(2) For any µ ∈ J , there exist at most two distinct covers of the form (α, µ) in

J , i.e., there are at most two elements in J covering µ.

(3) For any λ ∈ J , λ covers at most two distinct elements in J .

(4) If α, β are two covers of µ in J , then α∨β covers both α, β; thus the interval

[µ, α ∨ β] is a rank 2 subposet of J .


Example 4.3. The following is an example of a grid lattice.

4, 6EE

Eyy

y

3, 6EE

Eyy

y4, 5

yyy

2, 6EE

E3, 5

EEE

yyy

2, 5EE

E3, 4

EEE

yyy

2, 4EE

Eyy

y3, 3

yyy

1, 4EE

E2, 3

yyy

1, 3

1, 2

Definition 4.4. We now turn our attention to distributive lattices L such that

J(L) is a grid lattice. We will refer to such a lattice as a lattice with JIGL. (JIGL is

short for Join Irreducibles (forming a) Grid Lattice.)

As stated at the beginning of this chapter, our main goal is to study Bruhat-

Hibi toric varieties (defined in Definition 4.9). We now show the connection between

Bruhat-Hibi toric varieties and lattices with JIGL.

4.5. Minuscule weights and lattices. Let G be a semisimple, simply con-

nected algebraic group. Let T be a maximal torus in G. Let X (T ) be the character

group of T , and B a Borel subgroup containing T . Let R be the root system of G

relative to T ; let R+ (resp. S = {α1, · · · , αl}) be the set of positive (resp. simple)

roots in R relative to B (here, l is the rank of G). Let {ωi, 1 ≤ i ≤ l} be the funda-

mental weights. Let W be the Weyl group of G, and (, ) a W -invariant inner product

on X(T )⊗R. For generalities on semisimple algebraic groups, we refer the reader to

[4].


Let P be a maximal parabolic subgroup of G with ω as the associated fundamental

weight. Let WP be the Weyl group of P (note that WP is the subgroup of W generated

by {sα | α ∈ SP ⊂ S}). Let W P = W/WP . We have that the Schubert varieties of

G/P are indexed by W P , and thus W P can be given the partial order induced by the

inclusion of Schubert varieties.

Definition 4.6. A fundamental weight ω is called minuscule if 〈ω, β〉 (= 2(ω,β)(β,β)

) ≤

1 for all β ∈ R+; the maximal parabolic subgroup associated to ω is called a minuscule

parabolic subgroup.

Remark 4.7 (cf [15]). Let P be a maximal parabolic subgroup; if P is minuscule

then W/WP is a distributive lattice.

Definition 4.8. For P a minuscule parabolic subgroup, we call L = W/WP a

minuscule lattice.

Definition 4.9. We call X(L) a Bruhat-Hibi toric variety (B-H toric variety for

short) if L is a minuscule lattice.

For convenience, we list all of the minuscule fundamental weights here. Following

the indexing of the simple roots as in [5], we have the complete list of minuscule

weights for each type of semisimple algebraic group:

Type An : Every fundamental weight is minuscule

Type Bn : ωn

Type Cn : ω1

Type Dn : ω1, ωn−1, ωn

Type E6 : ω1, ω6

Type E7 : ω7.

There are no minuscule weights in types E8,F4, or G2.


Before proving that each minuscule lattice is a lattice with JIGL, we must intro-

duce some additional lattice notation. For a poset P , let I (P ) represent the lattice

of ideals of P . Thus for a distributive lattice L, L = I (J (L)) (cf. Theorem 2.9).

(Notice that the empty set is considered the minimal ideal, and in Theorem 2.9 we

do not include the minimal element in P . Therefore, in this section, I (J) will have

a minimal element that is not an element of J .)

For k ∈ N, let k be the the totally ordered set with k elements. The symbols ⊕

and × denote the disjoint union and (Cartesian) product of posets.

Let Xn (ωi) denote the minuscule lattice W/WP where P is a parabolic subgroup

associated to ωi in the root system of type Xn.

Theorem 4.10 (cf. [28], Propositions 3.2 and 4.11). The minuscule lattices have

the following combinatorial descriptions:

An−1 (ωj) ∼= I(I

(j − 1⊕ n− j − 1

))Cn (ω1) ∼= 2n

Bn (ωn) ∼= Dn+1 (ωn+1) ∼= Dn+1 (ωn) ∼= I (I (I (1⊕ n− 2)))

Dn (ω1) ∼= In−1 (1⊕ 1)

E6 (ω1) ∼= E6 (ω6) ∼= I4 (1⊕ 2)

E7 (ω7) ∼= I5 (1⊕ 2) .

This theorem is very convenient in working with the faces of B-H toric varieties,

because the join irreducible lattice of each of these minuscule lattices is very easy

to see, simply by eliminating one I (·) operation. Our goal is to show that the join

irreducibles of each minuscule lattice is in fact a grid lattice.

4.11. Minuscule lattices An−1 (ωj).

1Our notation differs significantly than that used in [28]; namely, where we use I, Proctor uses

J ; whereas we use J to signify the set of join irreducibles.


Remark 4.12 (cf. [28], Proposition 4.2). The join irreducibles of the minuscule

lattice An−1 (ωj) are isomorphic to the lattice

j × n− j.

Therefore, every element of J (An−1 (ωj)) can be written as the pair (a, b), for

1 ≤ a ≤ j, 1 ≤ b ≤ n− j. This leads us to the following result,

Corollary 4.13. The minuscule lattice An−1 (ωj) is a lattice with JIGL.

4.14. Minuscule lattices Cn (ω1). This minuscule lattice is totally ordered, and

the associated B-H toric variety is simply the affine space of dimension 2n.

4.15. Minuscule lattices Bn−1 (ωn−1) ∼= Dn (ωn−1) ∼= Dn (ωn). From Theorem

4.10, we have

J (Dn (ωn)) ∼= I2 (1⊕ n− 3) ∼= An−1 (ω2) .

It is a well known result that An−1 (ω2) represents the lattice of Schubert varieties

in the Grassmannian of 2-planes in Kn, and the Schubert varieties are indexed by

I2,n = {(i1, i2) | 1 ≤ i1 < i2 ≤ n}. Therefore,

J (Dn (ωn)) ∼= I2,n.

The lattice I2,n is therefore distributive, (being another minuscule lattice), and clearly

a grid lattice. This leads to the following result,

Corollary 4.16. The minuscule lattices Bn−1 (ωn−1) , Dn (ωn−1), and Dn (ωn)

are lattices with JIGL.


4.17. Minuscule lattices Dn (ω1). From Theorem 4.10, we have

J (Dn (ω1)) ∼= In−2 (1⊕ 1). This lattice of join irreducibles is isomorphic to the

following sublattice of N× N:

(n, n− 1)

(2, n− 1)

��

(2, n− 2)

ooooooo

(1, n− 1)

OOOOOOO

(1, n− 2)

ooooooo

OOOOOOO

(1, 1)

��

Clearly this is a grid lattice.

4.18. Minuscule lattices E6 (ω1) ∼= E6 (ω6), and E7 (ω7). Let E6 = E6 (ω1) =

E6 (ω6) and E7 = E7 (ω7). Since there are only two exceptional cases, it is best to

explicitly give the grid lattice structure to the join irreducibles. Thus, we have the

two join irreducible lattices below, with each lattice point given coordinates in N×N.

Coincidentally, J (E6) = D5 (ω5) and J (E7) = E6.


9, 9

8, 9

7, 9

6, 6 6, 9sss KK

K

5, 6 5, 9KKK

6, 8FFsss

4, 6KKKsss

5, 8KKKsss

6, 7xx

3, 6KKKxx

4, 5sss

4, 8KKKxx

5, 7sss

2, 6FF

3, 5KKKsss

3, 8FFxx

4, 7KKKsss

2, 5KKK

3, 4FFsss

2, 8FF

3, 7KKKxx

4, 6sss

2, 4KKKsss

3, 3xx

2, 7FF

3, 6KKKsss

1, 4KKK

2, 3sss

2, 6KKK

3, 5FFsss

1, 3 2, 5KKKsss

3, 4xx

1, 2 1, 5KKK

2, 4sss

1, 1 1, 4

1, 3

1, 2

1, 1

J (E6) J (E7)

This completes the individual discussion for each type of minuscule lattice, leading

us to the following result.

Corollary 4.19. If L is a minuscule lattice, then J (L) is a grid lattice.

5. RESULTS ON A LATTICE WITH JIGL 28

5. Results on a lattice with JIGL

For this entire section, we let L be a lattice with JIGL. Let J denote J(L). It will

often be useful to view elements of L as ideals in J . Recall that for x, y ∈ L, x ≥ y

if and only if Ix ⊇ Iy as ideals in J .

Lemma 5.1. Given γ1, γ2 ∈ J , (γ1 ∧ γ2)L belongs to J and is in fact equal to

(γ1 ∧ γ2)J .

Proof. Let θ = (γ1 ∧ γ2)J and φ = (γ1 ∧ γ2)L. Clearly θ ∈ Iγ1 ∩ Iγ2 = Iφ.

Therefore Iθ ⊂ Iφ. Let now η ∈ Iφ(⊂ J). Then η ≤ φ, and thus η is less than or

equal to both γ1 and γ2 in L, and therefore in J . Hence η ≤ θ, and thus Iφ ⊂ Iθ. The

result follows. �

Lemma 5.2. Let (α, β) be an incomparable pair of irreducibles (cf. Definition

2.6) in L. Then

(1) α, β are meet irreducibles in J ,

(2) (α ∧ β)L = (α ∧ β)J ∈ J .

Proof. Part (2) follows from Lemma 5.1, (note that α, β ∈ J). Now say α =

(γ1∧γ2)J for an incomparable pair (γ1, γ2) in J . Lemma 5.1 implies that α = (γ1∧γ2)L,

a contradiction since α is meet irreducible in L. Part (1) follows. �

Thus an incomparable pair (α, β) of irreducibles in L determines a (unique) non-

meet irreducible in J (namely, (α∧ β)L = (α∧ β)J). We shall now show (cf. Lemma

5.5 below) that conversely a non-meet irreducible element µ in J determines a unique

incomparable pair (α, β) of irreducibles in L. We first prove a couple of preliminary

results:

Lemma 5.3. Let µ be a non-meet irreducible element in J . Then µ determines an

incomparable pair (α, β) of elements (in J) both of which are meet irreducible in J .


Proof. Let µ = (µ1, µ2) (considered as an element of N×N). Since µ is non-meet

irreducible element in J , there exist x = (x1, x2), y = (y1, y2) in J , x, y > µ such that

x2 > µ2, y1 > µ1. Define α = (α1, α2), β = (β1, β2) in J as

α = the maximal element x > µ in J such that x1 = µ1,

β = the maximal element y > µ in J such that y2 = µ2.

Clearly α, β are both meet-irreducible in J (note that (µ1 + 1, α2) (resp. (β1, µ2 + 1))

is the unique element in J covering α (resp. β) in J). Also, it is clear that (α, β) is

an incomparable pair. �

Let µ, α, β be as in the above Lemma. In particular, we have, µ1 = α1 < β1, µ2 =

β2 < α2.

Lemma 5.4. With notation as in Lemma 5.3, we have,

(1) (α ∨ β)J = (β1, α2).

(2) α is the maximal element of the set {x = (x1, x2) ∈ J | x1 = α1}, and β is

the unique maximal element of the set {x = (x1, x2) ∈ J | x2 = β2}.

Proof. Assertion (2) is immediate from the definition of α, β. Assertion (1) is

also clear. �

Lemma 5.5. Let µ, α, β be as in Lemma 5.3. Then α and β are irreducibles in L.

Thus the non-meet irreducible element µ of J determines a unique incomparable pair

of irreducibles in L.

Proof. We will show the result for α (the proof for β being similar). Since

α ∈ J, α is join irreducible in L. It remains to show that α is meet irreducible in L.

If possible, let us assume that there exists an incomparable pair (θ1, θ2) in L such


that θ1∧θ2 = α; without loss of generality, we may suppose that θ1 and θ2 both cover

α. Then there exist (cf. Lemma 2.10) γ, δ ∈ J such that

Iθ1 = Iα∪̇{γ}, Iθ2 = Iα∪̇{δ}.

We have

Iγ ∩ Iδ ⊂ Iθ1 ∩ Iθ2 = Iα. (∗)

Also, γ, δ are either covers of α in J , or non-comparable to α. (They cannot be less

than α because they are not in Iα.)

Case 1: Suppose γ and δ are covers of α in J . Then α is not meet irreducible in

J , a contradiction (cf. Lemma 5.2,(1)).

Case 2: Suppose γ covers α in J , and δ is non-comparable to α. Let δ =

(δ1, δ2), ξ = (ξ1, ξ2) = (α∨δ)J . Then the fact that ξ > α (since α, δ are incomparable)

implies (in view of Lemma 5.4, (2)) that ξ1 > µ1; hence δ1(= ξ1) ≥ µ1+1, and δ2 < α2.

Also, γ = (µ1 + 1, α2) (cf. Lemma 5.4, (2)). Therefore γ ∧ δ = (µ1 + 1, δ2), but this

element is non-comparable to α, and thus Iγ ∩ Iδ 6⊂ Iα, a contradiction to (∗). Hence

we obtain that the possibility “γ covers α in J and δ is non-comparable to α” does

not exist. A similar proof shows that the possibility “δ covers α in J and γ is non-

comparable to α” does not exist.

Case 3: Suppose both γ = (γ1, γ2) and δ = (δ1, δ2) are non-comparable to

α = (µ1, α2). As in Case 2, we must have δ2 < α2, and thus δ1 > µ1. Similarly,

γ2 < α2, γ1 > µ1. Thus the minimum of {γ1, δ1} is still greater than µ1, therefore

Iγ ∩ Iδ 6⊂ Iα, a contradiction to (∗).

Thus our assumption that α is non-meet irreducible in L is wrong, and it follows

that α (and similarly β) is meet irreducible in L. �

We continue with the above notation; in particular, we denote µ = (µ1, µ2), µ1 =

α1 < β1, µ2 = β2 < α2.


Lemma 5.6. Let x = (x1, x2) ∈ J . If x 6∈ Iα ∪ Iβ, then x > α ∧ β.

Proof. By hypothesis, we have x 6≤ α, x 6≤ β.

We first claim that x1 > α1; for, if possible, let us assume x1 ≤ α1. Since x 6≤ α,

we must have x2 > α2. Thus x ∨ α = (α1, x2) (> α, since α 6≥ x); but this is a

contradiction, by the property of α (cf. Lemma 5.4,(2)). Hence our assumption is

wrong, and we get x1 > α1.

Similarly, we have, x2 > β2, and the result follows (note that by our notation (and

definition of α, β), we have α ∧ β = (α1, β2)). �

Definition 5.7. For an incomparable (unordered) pair (α, β) of irreducible ele-

ments in L, define

Lα,β = L \ [α ∧ β, α ∨ β].

Proposition 5.8. Lα,β is an embedded sublattice.

Proof. First, we show that Lα,β is a sublattice. To do this, we identify L with

the “lattice of ideals” of J . Thus, for x ∈ Lα,β, either Ix 6⊃ (Iα∩ Iβ) or Ix 6⊂ (Iα∪ Iβ),

by definition of Lα,β. Note that Iα ∩ Iβ = Iα∧β, and Iα ∪ Iβ = Iα∨β.

Case 1: Let x, y ∈ Lα,β such that Ix, Iy 6⊃ Iα∧β. Then clearly Ix ∩ Iy 6⊃ Iα∧β;

and thus x ∧ y ∈ Lα,β. We also have (by the definition of ideals) that α ∧ β 6∈ Ix, Iy

(note that α ∧ β ∈ J (cf. Lemma 5.2,(2))), therefore α ∧ β 6∈ Ix ∪ Iy, and therefore

x ∨ y ∈ Lα,β.

Case 2: Let x, y ∈ Lα,β such that Ix 6⊃ Iα∧β and Iy 6⊂ Iα∨β. Then clearly

Ix ∩ Iy 6⊃ Iα∧β and Ix ∪ Iy 6⊂ Iα ∪ Iβ. Hence, x ∨ y, x ∧ y are in Lα,β.

Case 3: Let x, y ∈ Lα,β such that Ix, Iy 6⊂ Iα∨β. Clearly Ix ∪ Iy 6⊂ Iα ∪ Iβ; hence,

x ∨ y ∈ Lα,β.

Claim: Ix ∩ Iy 6⊂ Iα ∪ Iβ.


Note that Claim implies that x∧y ∈ Lα,β. If possible, let us assume that Ix∩Iy ⊂

Iα∪Iβ. Now the hypothesis that Ix, Iy 6⊂ Iα∪Iβ implies that there exist θ, δ ∈ J such

that θ ∈ Ix, θ 6∈ Iα ∪ Iβ, and δ ∈ Iy, δ 6∈ Iα ∪ Iβ. Now Iθ ∩ Iδ ⊂ Ix ∩ Iy ⊂ Iα ∪ Iβ (note

that by our assumption, Ix ∩ Iy ⊂ Iα ∪ Iβ). Hence we obtain that either θ ∧ δ ≤ α or

θ ∧ δ ≤ β; let us suppose θ ∧ δ ≤ α (proof is similar if θ ∧ δ ≤ β). By Lemma 5.6, we

have that both θ, δ ≥ α ∧ β, and hence θ ∧ δ ≥ α ∧ β. Thus

α ≥ θ ∧ δ ≥ α ∧ β = (α1, β2) (∗∗)

Let ξ(= (ξ1, ξ2)) = θ ∧ δ. Then (∗∗) implies that ξ1 = α1; hence at least one of

{θ1, δ1}, say θ1 equals α1. This implies that θ2 > α2 (since θ 6∈ Iα). This contradicts

Lemma 5.4(2). Hence our assumption is wrong and it follows that Ix ∩ Iy 6⊂ Iα ∪ Iβ.

This completes the proof in Case 3. Thus we have shown that Lα,β is a sublattice.

Next, we will show that Lα,β is an embedded sublattice. Let x, y ∈ L be such

that x∨ y, x∧ y are in Lα,β. We need to show that x, y ∈ Lα,β. This is clear if either

x ∧ y 6≤ α ∨ β or x ∨ y 6≥ α ∧ β (in the former case, x, y 6≤ α ∨ β, and in the latter

case, x, y 6≥ α ∧ β). Let us then suppose that x ∧ y ≤ α ∨ β and x ∨ y ≥ α ∧ β; this

implies that x ∧ y 6≥ α ∧ β and x ∨ y 6≤ α ∨ β (since, x ∨ y, x ∧ y are in Lα,β), i.e.,

Ix ∩ Iy 6⊃ Iα ∩ Iβ and Ix ∪ Iy 6⊂ Iα ∪ Iβ. We will now show that x, y ∈ Lα,β.

Since α∧β 6∈ Ix∩ Iy, we have that one of the elements {x, y} must not be greater

than or equal to α ∧ β, say x 6≥ α ∧ β. This implies that x ∈ Lα,β. It remains to

show that y ∈ Lα,β. If y 6≥ α ∧ β, then we would obtain that y ∈ Lα,β. Let us then

assume that y ≥ α ∧ β; i.e. Iy ⊃ Iα ∩ Iβ. Note that for any δ ∈ Ix, we have δ ≤ x

and thus δ 6≥ α∧ β. By Lemma 5.6, δ ∈ Iα ∪ Iβ, and therefore Ix ⊂ Iα ∪ Iβ. Since by

hypothesis Ix ∪ Iy 6⊂ Iα ∪ Iβ, we must have Iy 6⊂ Iα ∪ Iβ. Therefore, y ∈ Lα,β.

This completes the proof of the assertion that Lα,β is an embedded sublattice,

and therefore the proof of the Proposition. �

6. SINGULAR LOCUS OF X(L) 33

6. Singular locus of X(L)

In this section, we determine the singular locus of X(L), L being a lattice with

JIGL. Let σ be the cone associated to X (L). We follow the notation of §1 and §2.

Definition 6.1. A face τ of σ is a singular (resp. non-singular) face if Pτ is a

singular (resp. non-singular) point of Xσ.

Definition 6.2. Let us denote by W the set of generators for σ as described in

Proposition 3.12. Let τ be a face of σ, and let Dτ be as in § 3.13. Define

W (τ) = {v ∈ W | fIα (v) = 0, ∀α ∈ Dτ}.

Then W (τ) gives a set of generators for τ .

6.3. Determination of W (τ). Let (α, β) be an incomparable (unordered) pair

of irreducible elements of L. By Proposition 5.8, Lα,β is an embedded sublattice

of L (Lα,β being as in Definition 5.7). Let τα,β be the face of σ corresponding to

Lα,β (cf. Lemma 3.14; note that Dτα,β = Lα,β). Let us denote τ = τα,β. Following

the notation of §4, let µ(= (µ1, µ2)) = α ∧ β, α1 = µ1, β2 = µ2. Since µ is not

meet irreducible in J , there are two elements A and B in J covering µ, namely,

A = (α1, β2 + 1), B = (α1 + 1, β2). Also, we have that A∨B (in the lattice J) covers

both A and B, (cf. Remark 4.2). Let C = (A ∨B)J ; then C = (α1 + 1, β2 + 1).

It will aid our proof below to notice a few facts about the generating set W (τ) of

τ . First of all, e1̂ is not a generator for any τα,β; because 1̂ ∈ Lα,β for all pairs (α, β),

and e1̂ is non-zero on fI1̂ .

Secondly, for any cover (y, y′), y > y′ in J (L), ey′ − ey is not a generator of τ if

y′ ∈ Lα,β, because fIy′ (ey′ − ey) 6= 0. Thus, in determining the elements of W (τ), we

need only be concerned with elements ey′ − ey of W such that y′ ∈ J ∩ [α∧ β, α∨ β].

Lemma 6.4. J ∩ [α ∧ β, α ∨ β] = {x ∈ J | x ∈ [µ, α] ∪ [µ, β]}.


Proof. The inclusion ⊇ is clear. To show the inclusion ⊆, let x ∈ J∩[α∧β, α∨β].

If possible, assume x 6∈ [µ, α] ∪ [µ, β]; the assumption implies that x 6∈ Iα ∪ Iβ(=

Iα∨β). Hence we obtain that x 6≤ α ∨ β, a contradiction to the hypothesis that

x ∈ [α ∧ β, α ∨ β]. �

Lemma 6.5. The set {x ∈ J | x 6∈ Iα ∪ Iβ} has a unique minimal element;

moreover that element is C.

Proof. For any x in this set, we have x > α ∧ β (cf. Lemma 5.6). Hence by

Lemma 5.4,(2), and the hypothesis that x 6∈ Iα∪ Iβ, we obtain that x1 > α1, x2 > β2.

Therefore,

{x ∈ J | x 6∈ Iα ∪ Iβ} = {x ∈ J | x1 > α1, x2 > β2}.

This set clearly has a minimal element, namely C = (α1 + 1, β2 + 1). �

Theorem 6.6. Following the notation from above, we have

W (τ) = {eµ − eA, eµ − eB, eA − eC , eB − eC}.

Proof. Claim 1: W (τ) ⊃ {eµ − eA, eµ − eB, eA − eC , eB − eC}.

We must show that for any x ∈ Lα,β, fIx is zero on these four elements of W . If

possible, let us assume that there exists a x ∈ Lα,β such that fIx is non-zero on some

of the above four elements. Then clearly x ≥ µ(= α ∧ β). Hence x 6≤ α ∨ β (since

x 6∈ [α ∧ β, α ∨ β]), i.e., Ix 6⊂ Iα ∪ Iβ. Therefore Ix contains some join irreducible γ

such that γ 6≤ α, β; hence, Iγ 6⊂ Iα ∪ Iβ. This implies (cf. Lemma 6.5) that γ ≥ C.

Hence we obtain that C ∈ Ix. Therefore, x ≥ C, and fIx is zero on all of the four

elements of Claim 1, a contradiction to our assumption. Hence our assumption is

wrong and Claim 1 follows.

Claim 2: W (τ) = {eµ − eA, eµ − eB, eA − eC , eB − eC}.


In view of § 6.3, it is enough to show that for all θ ∈ J ∩ [α ∧ β, α ∨ β], the

element eθ − eδ ∈ W which is different from the four elements of Claim 1 is not in

W (τ). In view of Lemma 6.4, it suffices to examine all covers in J of all elements

in ([µ, α] ∪ [µ, β])J . This diagram represents the part of the grid lattice J we are

concerned with:

α C ′′

A′′

{{{{{

AA

A

C ′

CCCCC

A′

||||||

BBBBBB

β

C

??

??

??

??

A

AA

AA

AA

AA

~~~~~

B

@@@@@

��

��

��

�

µ

~~~~~

@@@@@

In the diagram above, consider A′ = (α1, β2 + n), A′′ = (α1, β2 + n + 1), and

C ′ = (α1 + 1, β2 + n). Note that all elements of J ∩ [µ, α] can be written in the form

of A′. Thus, we need to check elements eA′ − eA′′ and eA′ − eC′ in W .

First, we observe that C ′ ∈ Lα,β, and fIC′ is non-zero on eA′ − eA′′ . Hence

eA′ − eA′′ 6∈ W (τ).

Next, let x = (A′ ∨ C)L, (note that x is not in J , and thus does not appear on

the diagram above). Then Ix = IA′ ∪ IC ; and we have x ∈ Lα,β (since, C 6∈ Iα ∪ Iβ

and x > C, we have, x 6≤ α ∨ β). Moreover, fIx is non-zero on eA′ − eC′ . Hence

eA′ − eC′ 6∈ W (τ).

This completes the proof for the interval [µ, α], and a similar discussion yields the

same result for the interval [µ, β].

Thus Claim 2 (and hence the Theorem) follows. �


As an immediate consequence of Theorem 6.6, we have the following

Theorem 6.7. Let (α, β) be an incomparable pair of irreducibles in L. We have an

identification of the (open) affine piece in X(L) corresponding to the face τα,β with the

product Z×(K∗)#J(L)−3, where Z is the cone over the quadric surface x1x4−x2x3 = 0

in P3.

Lemma 6.8. The dimension of the face τα,β equals 3.

Proof. By Theorem 6.6, a set of generators for τα,β is given by {eµ − eA, eµ −

eB, eA − eC , eB − eC}. We see that a subset of three of these generators is linearly

independent. Thus if the fourth generator can be put in terms of the first three, the

result follows. Notice that

(eµ − eA)− (eµ − eB) + (eA − eC) = eB − eC .

�

The following two lemmas hold for a general toric variety.

Lemma 6.9. Let Xτ be an affine toric variety with τ as the associated cone. Then

Xτ is a non-singular variety if and only if it is non-singular at the distinguished point

Pτ .

Proof. Only the implication⇐ requires a proof. Let then Pτ be a smooth point.

Let us assume (if possible) that Sing Xτ 6= ∅. We have the following facts:

(1) Sing Xτ is a closed T -stable subset of Xτ .

(2) Pτ ∈ Oθ, for every face θ of τ (see §1.4); in particular, Pτ ∈ Oθ, for some face

θ such that Pθ is a singular point, (such a θ exists, since by our assumption

Sing Xτ is non-empty).


Therefore we obtain that Pτ ∈ Sing Xτ , a contradiction. Hence our assumption is

wrong and the result follows. �

Lemma 6.10. Let τ be a face of σ. Then Pτ is a smooth point of Xσ if and only

if Pτ is a smooth point of Uτ , i.e., if and only if τ is generated by a part of a basis of

N (N is the Z dual of the character group of the torus).

Proof. Uτ is a principal open subset of Xσ. Hence Xσ is non-singular at Pτ if

and only if Uτ is non-singular at Pτ . By Lemma 6.9, Uτ is non-singular at Pτ if and

only if Uτ is a non-singular variety; but by §2.1 of [11], this is true if and only if τ is

generated by a part of a basis of N . �

Returning to our study of X(L), we combine the two lemmas above with Theorem

6.7 and we obtain the following:

Theorem 6.11. Pτ ∈SingXσ, for τ = τα,β. Further, the singularity at Pτ is of the

same type as that at the vertex of the cone over the quadric surface x1x4 − x2x3 = 0

in P3.

Next, we will show that the faces containing some τα,β are the only singular faces.

Lemma 6.12. Let (y, y′), y > y′ be a cover in J . Then either ey′ − ey ∈ W (τα,β)

for some incomparable pair (α, β) of irreducibles in L, or y, y′ are comparable to every

other element of J .

Proof. Case 1: Let y′ be non-meet irreducible in J .

In view of the hypothesis, we can find an incomparable pair (α, β) of irreducibles

in L such that y′ = α∧β, as shown in Lemma 5.5 (with µ = y′). Thus ey′−ey = eµ−eA

or ey′ − ey = eµ − eB as in Theorem 6.6.

Case 2: Let y′ be meet irreducible, but not join irreducible (in J).


Let x1 and x2 be the two elements covered by y′ in J (cf. Remark 4.2); thus

(x1 ∨ x2)J = y′.

For convenience of notation, all join and meet operations in this proof will refer

to the join and meet operations in the lattice J .

Claim (a): If both x1 and x2 are meet irreducible (in J), then y′, y are comparable

to every element of J .

If possible, let us assume that there exists a z ∈ J such that z is non-comparable

to y′. We first observe that z is non-comparable to both x1 and x2; for, say z, x1 are

comparable, then z > x1 necessarily (since z, y′ are non-comparable). This implies

that x1 ≤ z ∧ y′ < y′, and hence we obtain that x1 = z ∧ y′ < y′ (since (y′, x1) is a

cover), a contradiction to the hypothesis that x1 is meet irreducible. Thus we obtain

that z is non-comparable to both x1 and x2. Now, we have, z ∨ xi ≥ z ∨ y′ (note that

xi, i = 1, 2 being meet irreducible in J , y′ is the unique element covering xi, i = 1, 2,

and hence z ∨ xi ≥ y′). Hence (z ∨ y′ ≥)z ∨ xi ≥ z ∨ y′, and we obtain

z ∨ x1 = z ∨ y′ = z ∨ x2.

On the other hand, the fact that z ∧ y′ < y′ implies that z ∧ y′ ≤ x1 or x2. Let i be

such that z ∧ y′ ≤ xi. Then z ∧ y′ ≤ z ∧ xi ≤ z ∧ y′; therefore

z ∧ xi = z ∧ y′.

Now

y′ ∧ (xi ∨ z) = y′ ∧ (y′ ∨ z) = y′; (y′ ∧ xi) ∨ (y′ ∧ z) = xi ∨ (xi ∧ z) = xi.

Therefore J is not a distributive lattice (Definition 2.3), a contradiction. Hence our

assumption is wrong and it follows that y′ is comparable to every element of J , and

since y is the unique cover of y′, y is also comparable to every element of J . Claim

(a) follows.


Continuing with the proof in Case 2, in view of Claim (a), we may suppose that

x1 is not meet irreducible (in J). Then by Lemma 5.3 (with µ = x1), there exists a

unique incomparable pair (α, β) of meet irreducibles (in J) such that x1 = α ∧ β. In

view of the fact that y′ is a cover of x1, we obtain that y′ is equal to A or B (A, B

being as in §6.3), say y′ = A; this in turn implies that y = C (C being as in §6.3; note

that by hypothesis, y is the unique element covering y′ in J). Therefore we obtain

that ex1 − ey′(= ex1 − eA), ey′ − ey(= eA − eC) are in W (τα,β).

This completes the proof of the assertion in Case 2.

Case 3: Let y′ be both meet irreducible and join irreducible in J .

If y′ is comparable to every other element of J , then y is also comparable to every

other element of J , since by hypothesis, y is the unique element covering y′ in J ; and

the result follows.

Let then there exist a z ∈ J such that z and y′ are incomparable. This in particular

implies that y′ 6= 0̂J ; let x ∈ J be covered by y′ (in fact, by hypothesis, x is unique).

Proceeding as in Case 2 (especially, the proof of Claim (a)), we obtain that x is non-

meet irreducible. Hence taking µ = x in Lemma 5.3 and proceeding as in Case 2, we

obtain that ey′ − ey is in W (τα,β) ((α, β) being the incomparable pair of irreducibles

determined by µ).

This completes the proof of the Lemma. �

Theorem 6.13. Let τ be a face of σ such that Dτ is not contained in any Lα,β,

for all incomparable pair (α, β) of irreducibles in L; in other words τ does not contain

any τα,β. Then τ is nonsingular.

Proof. As in Definition 6.2, let

W (τ) = {v ∈ W | fIα (v) = 0, ∀α ∈ Dτ}.


Then W (τ) gives a set of generators for τ . By Remark 6.10 and §2.1 of [11], for τ to

be nonsingular, it must be generated by part of a basis for N (N being as in § 1.2). If

W (τ) is linearly independent, then it would follow that τ is non-singular. (Generally

this is not enough to prove that τ is nonsingular; but since all generators in W have

coefficients equal to ±1, any linearly independent subset of W will serve as part of a

basis for N .)

If possible, let us assume that W (τ) is linearly dependent. Recall that the elements

of W can be represented as all the line segments in the lattice J , with the exception of

e1̂. Therefore, the linearly dependent generators W (τ) of τ must represent a “loop”

of line segments in J . This loop will have at least one bottom corner, left corner, top

corner, and right corner.

Let us fix an incomparable pair (α, β) of irreducibles in L. By Theorem 6.6, we

have that W (τα,β) = {eµ − eA, eµ − eB, eA − eC , eB − eC} (notation being as in that

Theorem). These four generators are represented by the four sides of a diamond in J .

Thus, by hypothesis, the generators of τ represent a loop in J that does not traverse

all four sides of the diamond representing all four generators of τα,β. We have the

following identification for Lα,β:

Lα,β = {x ∈ L | fIx ≡ 0 on W (τα,β)}. (†)

The above identification for Lα,β together with the hypothesis that Dτ 6⊆ Lα,β implies

the existence of a θ ∈ Dτ ∩ [α ∧ β, α ∨ β]; note that by (†), we have

fIθ 6≡ 0 on W (τα,β)

This implies in particular that θ 6≥ C (C being as the proof of Theorem 6.6); also,

θ ≥ µ(= α ∧ β), since θ ∈ [α ∧ β, α ∨ β]. Based on how θ compares to both A and

B, we can eliminate certain elements of W from W (τ). There are four possibilities;

we list all four, as well as the corresponding generators in W (τα,β) which are not in


W (τ), i.e., those generators v in W (τα,β) such that fIθ(v) 6= 0:

θ 6≥ A, θ 6≥ B ⇒ eµ − eA, eµ − eB 6∈ W (τ)

θ ≥ A, θ 6≥ B ⇒ eA − eC , eµ − eB 6∈ W (τ)

θ 6≥ A, θ ≥ B ⇒ eµ − eA, eB − eC 6∈ W (τ)

θ ≥ A, θ ≥ B ⇒ eA − eC , eB − eC 6∈ W (τ)

Therefore, we obtain

neither {eµ − eA, eA − eC} nor {eµ − eB, eB − eC} is contained in W (τ) (∗)

for any τα,β ((α, β) being an incomparable pair of irreducibles in L).

Let y′, z′ denote respectively, the left and right corners of our loop; let (y, y′), (z, z′)

denote the corresponding covers (in J) which are contained in our loop. Now y′, z′

are non-comparable; hence, by Lemma 6.12 we obtain that (y, y′) (resp. (z, z′)) are

contained in some W (τα,β) (resp. W (τα′,β′)). Hence we obtain (by Theorem 6.6, with

notation as in that Theorem)

{eµ − ey′ , ey′ − ey} = {eµ − eA, eA − eC} or {eµ − eB, eB − eC}

But this contradicts (∗). Thus our loop in J that represented W (τ) cannot have both

left and right corners; therefore W (τ) is not a loop at all, a contradiction. Hence, our

assumption (that W (τ) is linearly dependent) is wrong, and the result follows. �

Combining the above Theorem with Theorem 6.11 and Lemma 6.8, we obtain our

first main Theorem:

Theorem 6.14. Let L be a distributive lattice such that J (L) is a grid lattice.

Then

7. DIVISORS AND LINE BUNDLES 42

(1) Sing X (L) =⋃

(α,β)

Oτα,β , the union being taken over all incomparable pairs

(α, β) of irreducibles in L.

(2) SingX (L) is pure of codimension 3 in X (L); further, the generic singulari-

ties are of cone type (more precisely, the singularity type is the same as that

at the vertex of the cone over the quadric surface x1x4 − x2x3 = 0 in P3 ).

As a result of Corollary 4.19, we have the following, letting

Φ = {(α, β) | α, β non-comparable irreducibles in L},

Theorem 6.15. For the B-H toric variety X (L),

Sing X (L) =⋃

(α,β)∈Φ

Oτα,β .

In other words, X (L) is smooth at Pτ (τ being a face of σ) if and only if for each pair

(α, β) ∈ Φ, there exists at least one γ ∈ [α ∧ β, α ∨ β] such that Pτ (γ) is non-zero.

7. Divisors and line bundles

7.1. Weil divisors.

Definition 7.2. For any variety X, a Weil divisor is an element of the free

abelian group generated by irreducible closed subvarieties of codimension one in X.

If X is a toric variety, a Weil divisor that is mapped to itself by the torus T is a

T -Weil divisor.

Thus for Xσ where σ is a convex polyhedral cone, let τ1, . . . , τd be the one-

dimensional faces of σ. Then the orbit closures Oτ1 , . . . , Oτd are all of the T -stable

irreducible subvarieties of codimension one. Thus, T -Weil divisors are of the form∑di=1 aiOτi .


Now let X(L) be a Hibi toric variety. Due to our (inclusion-reversing) correspon-

dence between faces of the cone and embedded sublattices (see §3.13), we also have

a correspondence between one-dimensional faces and maximal embedded sublattices.

Here we give a method for finding these maximal embedded sublattices.

Let the elements of J(L) generate N = Z#J(L). Let α2 cover α1 in J(L), so we

have a one-dimensional face generated by eα1 − eα2 (using notation as in Proposition

3.12). We define the following subset of L:

Lα2α1 = L \ {a ∈ L | a ≥ α1 and a 6≥ α2}.

Clearly, Lα2α1 is the maximal subset of L with the property that for a ∈ Lα2α1

, fIa

vanishes on eα1 − eα2 .

Lemma 7.3. Lα2α1 is an embedded sublattice.

Proof. First we show that Lα2α1 is in fact a sublattice. Let b, c ∈ Lα2α1

. If both

b, c ≥ α1, then we have b, c ≥ α2, and thus b∨c, b∧c ≥ α2. Therefore b∨c, b∧c ∈ Lα2α1 .

Now say b, c 6≥ α1. Viewing L as the lattice of ideals of J(L), this implies α1 6∈ Ia∪Ib.

Thus a ∨ b 6≥ α1. Clearly a ∧ b 6≥ α1, therefore a ∧ b, a ∨ b ∈ Lα2α1 . Lastly, say

a ≥ α2, b 6≥ α1, but then clearly a ∨ b ≥ α2, and a ∧ b 6≥ α1. Thus we have shown

that Lα2α1 is a sublattice of L.

Next we show that Lα2α1 is an embedded sublattice. Let a, b ∈ L such that a∨b, a∧

b ∈ Lα2α1 . We’ll show that for a ∨ b ≥ α2, a ∧ b 6≥ α1 we must have a, b ∈ Lα2α1

. (The

other cases follow obviously.) Since α2 ∈ Ia ∪ Ib, we must have one of a, b ≥ α2, say

a. Conversely, we have α1 6∈ Ia∩ Ib. Clearly α1 ∈ Ia, we must have b 6≥ α1. Therefore

a, b ∈ Lα2α1 . �

We have that X(Lα2α1) is an irreducible closed subvariety of codimension one in

X(L) (from Proposition 3.15).


In addition, for any α ∈ J(L) such that α is a maximal element, then we have a

one-dimensional face generated by eα. The associated embedded sublattice is Lα =

{a ∈ L | a 6≥ α}. (The proof that this is in fact an embedded sublattice follows very

easily from the fact that α is both join-irreducible, and maximal.)

For a toric variety X(L) where L is a lattice with JIGL, we have a unique maximal

element of J(L), say 1̂. Therefore, T -Weil divisors are of the form

D = a1̂X(L1̂) +∑

(αj ,αi)∈Z(J(L))

ajiX(Lαjαi ).

(Z(J(L)) is the set of all covers in J(L), as in Proposition 3.12.)

7.4. Cartier divisors. From [11, §3.3], a Cartier divisor for a variety X is given

by the data of a covering of X by affine open sets Uα, and nonzero rational functions

fα called local equations, such that the ratios fα/fβ are nowhere zero regular functions

on Uα ∩ Uβ. A Cartier divisor D determines a Weil divisor, denoted [D], by

[D] =∑

codim(V,X)=1

ordV (D) · V,

where ordV (D) is the order of vanishing of an equation for D in the local ring along the

subvariety V . For a normal variety X, the map D 7→ [D] embeds the group of Cartier

divisors in the group of Weil divisors. A nonzero rational function f determines a

principal divisor div(f) whose local equation in each open set is f .

Cartier divisors which are T -stable are called T -Cartier divisors. Let σ be a convex

polyhedral cone, and u ∈ σ̌∩M (note that we are using notation introduced in §1.2).

Let x ∈ Xσ, then x can be realized as a semigroup homomorphism from σ̌ ∩M to C.

The function χu is defined such that χu(x) = x(u).


Lemma 7.5 ([11]). A general T -Cartier divisor on Xσ has the form div(χu) for

some u ∈M . Moreover, the associated Weil divisor is given by

[div(χu)] =∑

i

〈u, vi〉Di

where vi is the first lattice point along the one-dimensional face τi, 〈 , 〉 is the canonical

pairing on N ×M , and Di = Oτi.

Now let X(L) be a Hibi toric variety, where L is a lattice with JIGL. It is a

straight forward application of the lemma above to see that T -Cartier divisors on

X(L) are of the form

〈u, e1̂〉X(L1̂) +∑


〈u, eαi − eαj

〉X(Lαjαi ).

For u = (u1, . . . , un) ∈ Zn where n is the cardinality of J(L), the sum above is equal

to

unX(L1̂) +∑


(ui − uj) X(Lαjαi ).

Example 7.6. Let us consider a basic grid lattice:

α4

J(L) = α2

}}}}}α3

AAAAA

α1

}}}}}

AAAAA

Let Dji = X(Lαjαi ). Thus, a T -Weil divisor on X(L) is of the form f = xD21 + yD31 +

zD42 + wD43, for integers x, y, z, w (for now, we ignore the term corresponding to L1̂,

since it has no relations with the other subvarieties). If f is a T -Cartier divisor, then


there exists a u = (u1, u2, u3, u4) ∈ Z4 such that

u1 − u2 = x

u1 − u3 = y

u2 − u4 = z

u3 − u4 = w

We have that, for f to be a Cartier divisor, x− y + z − w = 0.

Now, for J(L) any grid lattice, a Cartier divisor has a relation like the one above

for every rank 2 diamond in the grid lattice. All other relations are a result of these.

7.7. Line bundles. For a Cartier divisor D, the ideal sheaf O(−D) is the sub-

sheaf of the sheaf of rational functions generated by fα on Uα; the inverse sheaf

O(D) is the subsheaf of OX generated by 1/fα on Uα. Regarded as a line bundle, its

transition functions from Uα to Uβ are fα/fβ.

Let Pic(X) be the group of all line bundles, up to isomorphism. As a result

of [11, §3.4], we have that rank(Pic(X)) ≤ d − n, where d is the number of one-

dimensional faces in σ and n is the dimension of the variety. Thus, for X(L) a Hibi

toric variety where J(L) is a grid lattice, we have rank(Pic(X(L))) ≤ #{covers in

J(L)} + 1 − #J(L). (Note that the “+1” comes from the one maximal element in

J(L).)

Proposition 7.8. Let L be a lattice with JIGL, then rank(Pic(X(L))) is bounded

by the number of minimal diamonds in J(L).

Proof. First, it is easily checked that the proposition holds for the two most

basic grid lattices: first, if J(L) is a totally ordered chain of n + 1 elements, then

there are n covers. Hence, from the statement above, we have rank(Pic(X(L))) ≤


n + 1 − (n + 1) = 0. Since there are no diamonds in a totally ordered chain, the

proposition holds.

The second grid lattice we will check: let J(L) be four elements in a minimal

diamond, as in Example 7.6. Now we have rank(Pic(X(L))) ≤ 4 + 1 − 4 = 1. Once

again, the proposition holds.

Finally, we let J(L) be any grid lattice. Say that J(L) = J1∪J2, where J1 and J2

are grid lattices, such that J1 and J2 intersect on some totally ordered interval of n+1

elements. Thus the intersection also includes n covers. (Note that it is impossible for

the union of J1 and J2 to create new minimal diamonds in J(L) that did not already

exist in J1 or J2.) Let vi denote the number of lattice points in Ji, and ei denote the

number of cover in Ji.

Therefore, we have a bound on rank(Pic(X(L))); namely

rank(Pic(X(L))) ≤ (e1 + e2 − n) + 1− (v1 + v2 − (n + 1))

= e1 + e2 + 2− v1 − v2

= (e1 + 1− v1) + (e2 + 1− v2)

We can see that the bound for rank(Pic(X(L))) is actually the sum of the bounds on

the lattices corresponding to J1 and J2.

For any grid lattice J(L) we have J(L) =⋃

Ji, where each Ji is either a minimal

diamond or a totally ordered chain, and where the intersection of any pair Ji ∩ Jj is

either empty or a chain. The proposition follows. �

A T -Cartier divisor D =∑

aiDi determines a rational convex polyhedron in MR

defined by

PD = {u ∈MR | 〈u, vi〉 ≥ −ai for all i}

8. YOUNG LATTICES 48

where vi is the first lattice point on the one-dimensional face τi (τi corresponding to

Di).

Lemma 7.9 ([11]). The global sections of the line bundle O(D) are

Γ(X,O(D)) =⊕

u∈PD∩M

C · χu.

Now let X(L) be a Hibi toric variety, where L is a lattice with JIGL. Let D

be a T -Cartier divisor, D =div(χu) = unX(L1̂) +∑

(αj ,αi)(ui − uj) X(L

αjαi ) for some

u = (u1, . . . un). Therefore

PD = {v ∈MR | vn ≥ −un, vi − vj ≥ uj − ui for all covers (αj, αi) in J(L)}.

Let us return to scenario of Example 7.6. The system of inequalities for v ∈ PD

is:

v4 ≥ −u4

v1 − v2 ≥ u2 − u1

v1 − v3 ≥ u3 − u1

v2 − v4 ≥ u4 − u2

v3 − v4 ≥ u4 − u3

Thus, vi ≥ −ui for 1 ≤ i ≤ 4. It is easy to see that this result can be expanded to

any grid lattice. Therefore, for a T -Cartier divisor D =div(χu), u ∈M , we have

PD = {v ∈MR | vi ≥ −ui ∀ i}.

8. Young lattices

In this section, we show that a Young lattice L is a lattice with JIGL. (We give

the definition of a Young lattice below.) In [12, §10], it was shown that the multicone


over a partial flag variety SLn/Q flatly degenerates to the toric variety X(L) for L a

Young lattice. Thus, we will be showing that Theorems 6.14, 6.15 hold for this toric

degeneration of SLn/Q, for Q a parabolic subgroup.

For Q some maximal parabolic subgroup, we have Q =⋂r

i=1 Pdi where Pdi is the

maximal parabolic subgroup corresponding to fundamental weight ωdi , and 1 ≤ d1 <

. . . < dr ≤ n− 1. We define the distributive lattice

HQ =⋃̇r

i=1Idi,n.

Here, Id,n = {i = (i1, . . . , id) | 1 ≤ i1 < i2 < . . . < id ≤ n}, with a partial order on

Id,n given by i ≤ j if and only if i1 ≤ j1, . . . , id ≤ jd. For an element τ ∈ HQ ∩ Idt,n,

we say τ is of type dt. The partial order on two elements of HQ of type dt is given

by the partial order on Idt,n. For τ = (τ1, . . . , τdt), θ = (θ1, . . . , θds) of type dt, ds

respectively, we define τ ≤ θ if and only if

dt ≥ ds, and τ1 ≤ θ1, . . . , τds ≤ θds .

HQ is referred to as a Young lattice.

Theorem 8.1 (cf. [12]). The multicone over SLn/Q flatly degenerates to X(HQ).

A note about notation, we will use [i, j] to denote the segment i, i + 1, . . . , j, and

if there is a possibility that j < i, the segment is understood to be empty.

In the remainder of the section, we will show that HQ is a lattice with JIGL. We

begin by identifying J(HQ). Note that if both θ, δ ∈ HQ are of type d, then θ∨δ is of

type d. Thus, if an element in HQ of type d is not join irreducible in Id,n, it will not

be join irreducible in HQ. Thus, when identifying join irreducible elements in HQ, we

need only consider join irreducible elements in Idi,n for each i, 1 ≤ i ≤ r. From [13,

Lemma 8.2], we have that i ∈ Id,n is join irreducible if and only if i consists entirely


of one segment, or if i is equal to two disjoint segments, (µ, ν) such that µ starts with

1.

Let θ ∈ HQ be of type di and δ ∈ HQ be of type dj, such that di < dj. Then we

have θ ∨ δ = (max{θ1, δ1}, . . . , max{θdi , δdi}). Note that θ ∨ δ is of type di.

Let τ ∈ HQ be of type dt, τ = ([1, s], [p, p + dt − s− 1]), where 0 ≤ s ≤ dt,

p > n− dt+1 + s + 1. If t = r (i.e., dt+1 does not exist), then there is no condition on

p other than p > s.

Lemma 8.2. The element τ as defined above is in J(HQ).

Proof. Assume, if possible, that there exists θ, δ ∈ HQ distinct from τ such that

θ ∨ δ = τ . From the discussion above, we can see that τ is join irreducible as an

element of Idt,n, thus we must have θ and δ of different types in HQ, and one must be

of the same type as τ . Therefore, without loss of generality, let θ be of type dt and

δ of type dq where dt < dq; thus θ = (θ1, . . . , θdt), δ = (δ1, . . . , δdq). (Note that, by

this argument, we can already see that the lemma is true when τ is type dr, so we

continue assuming t < r.)

Since τ = θ ∨ δ, we have θi, δi ≤ τi for 1 ≤ i ≤ dt. Therefore (θ1, . . . , θs) =

(1, . . . , s) = (δ1, . . . , δs). We now examine {δs+1, . . . , δq}. By necessity, we have

δs+1 ≤ n−dq +s+1 (for δdq ≤ n), and since dq ≥ dt+1, we have δs+1 ≤ n−dt+1+s+1.

From the definition of τ , τs+1 = p > n − dt+1 + s + 1, therefore δs+1 < p and thus

θs+1 = p. We can continue this argument for each element of {δs+1, . . . , δdq}, until we

conclude that θ = τ . This is a contradiction, therefore τ must be join irreducible. �

Our next lemma will show that J(HQ) consists only of elements like τ above. As

discussed, it is enough to show that for an element ζ ∈ J(Idt,n), ζ is in J(HQ) if and

only if ζ can be written as τ above. Let ζ = ([1, s], [ζs+1, ζs+1 + dt − s− 1]), where


0 ≤ s < dt, and s + 1 < ζs+1 ≤ n − dt+1 + s + 1. We are assuming that ζ is of type

dt 6= dr. (All join irreducible elements of Idr,n are in J(HQ).)

Lemma 8.3. The element ζ as defined above is not join irreducible.

Proof. Let θ = (1, . . . , dt), and δ = ([1, s], [ζs+1, ζs+1 +dt+1−s−1]). Note that δ

is of type dt1 , and by our conditions on ζs+1, we have ζs+1 +dt+1−s−1 ≤ n, therefore

δ is a valid element of HQ. Clearly θ ∨ δ = ζ, and the result follows. �

Corollary 8.4. The elements of J(HQ) of type dt are of the form

τ = (1, . . . , s, p, . . . , p + dt + s− 1) , 0 ≤ s ≤ dt, p > n− dt+1 + s + 1,

(with condition on p only if dt < dr).

Now we will show that J(HQ) is a grid lattice. First, we see that J(HQ)dt = {τ ∈J(HQ) | τ of type dt} is independently a grid lattice for each dt, 1 ≤ t ≤ r. We knowthat J(HQ)dr = J(Idr,n) is a grid lattice because Idr,n is a minuscule lattice. Now

take dt 6= dr. To make notation easier, let ct = n − dt+1 + dt. Here we give J(HQ)dt(drawn horizontally):

[1, dt − 1], n ___________________ [n− dt + 1, n]

[1, dt − 1], ct + 2

��

[1, dt − 2], ct + 1, ct + 2 _________

��

[1, dt] [1, dt − 1], ct + 1 [1, dt − 2], ct, ct + 1 _____ [n− dt+1 + 2, ct + 1]

��

We have an identification of J(HQ)dt with a subset of Z×Z by sending ([1, dt]) to the

point (0, 1), and sending ([1, dt − a], [ct + b− a, ct + b]) to the point (a, b). Thus, as a

sublattice of Z×Z, J(HQ)dt is isomorphic to the interval from (1, 1) to (dt, dt+1− dt)

in union with the point (0, 1).

The only thing remaining to show is that the union⋃r

i=1 J(HQ)di is still a grid

lattice. This fact will be made clear by examining the union J(HQ)dt ∪ J(HQ)dt−1 .

9. A COUNTER EXAMPLE 52

The partial order relations on this union can be seen by examining the intervals

[(1, . . . , dt − 1, n) , (n− dt + 1, . . . , n)] ⊂ J(HQ)dt and[(1, . . . , dt−1) , (n− dt + 2, . . . , n− dt + dt−1 + 1)] ⊂ J(HQ)dt−1 .

[1, dt−1] _______ 1, [n− dt + 3, ct−1 + 1] [n− dt + 2, ct−1 + 1]

[1, dt − 1], n __ [1, dt−1], [ct−1 + 1, n] ______ 1, [n− dt + 2, n] [n− dt + 1, n]

From this, it is clear that the union of these two grid lattices forms a grid lattice.

Since J(HQ) can be constructed as a series of these unions (one on top of another),

the main result of the section is clear.

Theorem 8.5. The Young Lattice HQ =⋃r

i=1 Idi,n, 1 ≤ d1 < d2 < . . . < dr ≤ n−1

is a lattice with JIGL.

Example 8.6. Let SLn/Q = SL5/B, where B =⋂4

i=1 Pi. Thus, (drawn horizon-

tally) J(HQ) =

(1) (5)

(1, 2) (1, 5) (4, 5)

(1, 2, 3) (1, 2, 5) (1, 4, 5) (3, 4, 5)

(1, 2, 3, 4) (1, 2, 3, 5) (1, 2, 4, 5) (1, 3, 4, 5) (2, 3, 4, 5)

9. A counter example

It seems natural to ask whether or not the methods and results from the previous

sections (specifically Theorem 6.14) apply to a larger class of Hibi toric varieties than

just those based upon a lattice with JIGL. We provide the following example to show

that in fact, neither the methods nor results apply.


Let L be a distributive lattice. If J(L) is a sublattice of Nn, then we call J(L) ann-grid lattice. For example, we let L be a lattice such that J(L) is a cube.

H

E

wwwF G

GGG

B

wwwC

GGG wwwD

GGG

A

GGG www

J(L)

H

E ∨ F ∨G

E ∨ F

rrrrrrrrrrE ∨G F ∨G

SSSSSSSSSSSSSSS

E ∨D

rrrrrrrrrrF ∨ C

MMMMMMMMMM

kkkkkkkkkkkkkkkkG ∨B

SSSSSSSSSSSSSSSS

E

��F

��B ∨ C ∨D

UUUUUUUUUUUUUUUUUU

HHHHHHHHH

zzzzzzzzG

777777

B ∨ C

777777

iiiiiiiiiiiiiiiiiiB ∨D

>>>>>>>

vvvvvvvvvC ∨D

DDDDDDDD

��

B

rrrrrrrrrrrC

MMMMMMMMMMM

kkkkkkkkkkkkkkkkkkD

SSSSSSSSSSSSSSSSS

A

MMMMMMMMMMMMM

kkkkkkkkkkkkkkkkkkk

L

First, we note that taking a skew pair of join-meet irreducibles α, β and setting

Lα,β = L \ [α ∧ β, α ∨ β]


does not necessarily give us an embedded sublattice. In the example above, let our

skew pair be E, G. Clearly E ∧G = C and thus LE,G is the following sublattice:

H

E ∨ F ∨G

E ∨ F

iiiiiiiiF ∨G

TTTTTTT

F ∨ C

UUUUUUUUUUjjjjjjjjj

F

rrrrr

B ∨D

LLLLL

B

iiiiiiiiiiiiD

TTTTTTTTTT

A

UUUUUUUUUUUUUU

jjjjjjjjjjjj

Here, we have that F ∨(B∨C∨D), F ∧(B∨C∨D) are in LE,G, however, the element

B ∨ C ∨ D is not LE,G. Therefore, the sublattice LE,G is not embedded. Since the

sublattice is not embedded, it does not coincide with a face of the cone associated to

the toric variety X(L). Thus, we can assert that Theorem 6.15 cannot be extended

to this generalization of a lattice with join irreducibles forming an n-grid lattice.

Moreover, the singular locus in this example is not “pure” of codimension 3.

To illustrate this, we first note that the singular locus does have a codimension 3

piece similar to the JIGL case (see Theorem 6.6). The sublattice of L given by the

interval [E, H] corresponds to the face of the cone with generators {eA − eB, eA −

eC , eB − eE, eC − eE}. On the other hand, we note that the singular locus also has

a codimension 5 piece not contained in any codimension 3 piece. The sublattice of L

given by the three elements {A, E ∨ F ∨ G, H} corresponds to the face of the cone

with six generators {eB − eE, eB − eF , eC − eE, eC − eG, eD− eF , eD− eG}. We have

an identification of the (open) affine piece in X(L) corresponding to this face with

10. MULTIPLICITY FORMULAE FOR G-H TORIC VARIETIES 55

the product D2 × (K∗)3, where D2 is the determinantal variety with defining ideal

given by all 2-minors on a 3× 3 generic matrix.

This example, and other examples in which J(L) is an n-grid lattice have led us

to the following conjecture.

Conjecture 9.1. Let L be a distributive lattice such that J(L) is an n-grid

lattice. Then the singular locus of X(L) has irreducible components of codimension

3, 5, . . . , and 2n− 1.

10. Multiplicity formulae for G-H toric varieties

In this section, we restrict our attention to a specific class of B-H toric varieties:

specifically X(L) for L the Weyl group of the traditional Grassmannian. It is a well

known result (cf. [2]) that

L = Id,n = {i = (i1, . . . , id) | 1 ≤ i1 < i2 < . . . < id ≤ n} for 1 ≤ d < n.

We will sometimes use the notation Xd,n to denote X(Id,n), and refer to Xd,n as a

Grassmann-Hibi toric variety, or G-H toric variety, for short.

For 1 ≤ i ≤ n− d− 1, 1 ≤ j ≤ d− 1, let

µij = (1, . . . j, i + j + 1, . . . i + d) , and

λij = (i + 1, . . . i + j, n + 1 + j − d, . . . n) .

Define

Lij = L\ [µij, λij].

Proposition 10.1. From [13, §8], we have the following facts about the distrib-

utive lattice Id,n:


(1) The element τ = (i1, . . . , id) is irreducible if and only if either τ is a segment,

or τ consists of two disjoint segments (µ, ν), with µ starting with 1 and ν

ending with n.

(2) For any incomparable pair of irreducibles (α, β), there exists 1 ≤ i ≤ n−d−1,

1 ≤ j ≤ d− 1 such that α ∨ β = λij, α ∧ β = µij.

Thus, we have that Lij defined above is an embedded sublattice, playing the role

of Lα,β from Definition 5.7. Let σi,j denote the singular face of σ corresponding to

Lij.

10.2. Multiplicities of singular faces of X2,n. In this section, we take L =

I2,n, determine the multiplicity of X2,n at Pτ for certain of the singular faces of X2,n,

and deduce a product formula. Above we defined Lij and the corresponding face σi,j

for 1 ≤ j ≤ d− 1, 1 ≤ i ≤ n− d− 1; thus for I2,n, we need to consider only Li,1 for

1 ≤ i ≤ n− 3.

Example 10.3. Below is the poset of join irreducibles for I2,6. We write σi,1 inside

each diamond because the four segments surrounding it represent the four generators

of the face.

(5, 6)

(1, 6)

uuuuσ3,1 (4, 5)

IIII

(1, 5)

IIII uuuuσ2,1 (3, 4)

IIII

(1, 4)

IIII uuuuσ1,1 (2, 3)

IIII

(1, 3)

IIII uuuu

(1, 2)


To go from the join irreducibles of I2,6 to I2,7, we just add (1, 7) and (6, 7) to the poset

above, forming σ4,1. We will see that this makes the calculation of the multiplicities

of singular faces of I2,n much easier.

In the sequel, we shall denote the set of join irreducibles of I2,n by J2,n; also, as

in the previous sections, σ will denote the polyhedral cone corresponding to X2,n.

10.4. MultPσ X2,n. Now Xd,n being of cone type (i.e., the vanishing ideal is ho-

mogeneous), we have a canonical identification of TPσXd,n (the tangent cone to Xd,n

at Pσ) with Xd,n. Hence by Theorem 3.8, we have that multPσ Xd,n equals the number

of maximal chains in Id,n. So we begin by counting the number of maximal chains in

I2,n.

As we move through a chain from (1, 2), at any point (i, j) we have at most two

possibilities for the next point: (i + 1, j) or (i, j + 1). For each cover in our chain,

we assign a value: for a cover of type ((i, j + 1) , (i, j)) assign +1; for a cover of type

((i + 1, j) , (i, j)) assign −1.

A maximal chain C in I2,n contains 2n − 3 lattice points, and thus every chain

can be uniquely represented by a (2n − 4)-tuple of 1’s and −1’s; let us denote this

(2n− 4)-tuple by nC = 〈a1, . . . , a2n−4〉.

Of course we have restrictions on nC . First it is clear that 1 and −1 occur precisely

n − 2 times. Secondly, we can see that a1 = +1, and for any 1 ≤ k ≤ 2n − 4, if

{a1, . . . , ak} contains more −1’s than +1’s, then we have arrived at a point (i, j) with

i > j, which is not a lattice point. Thus, we must have a1 + . . . + ak ≥ 0 for every

1 ≤ k ≤ 2n− 4.

Theorem 10.5 (cf. Corollary 6.2.3 in [32]). The Catalan number

Catn =1

n + 1

2nn

, (n ≥ 0)

10. MULTIPLICITY

Documents

Some geometric properties of certain toric varieties and ...1577/fulltext.pdftoric variety. 7. 1. BRUHAT-HIBI TORIC VARIETIES 8 For Lbeing the Bruhat poset of Schubert varieties in